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This is a lecture notes take from the following link on numerical calculation of atomic physics:http://www.phys.ubbcluj.ro/~lnagy/pdf/1curs.pdf

This is a lecture notes take from the following link on numerical calculation of atomic physics:http://www.phys.ubbcluj.ro/~lnagy/pdf/1curs.pdf

I am trying to evaluate the two electron matrix element between $\langle2p_{1}(\vec{r}_{1})2p_{-1}(\vec{r}_{2})|\frac{1}{|\vec{r}_{1}-\vec{r}_{2}|}|2p_{1}(\vec{r}_{1})2p_{-1}(\vec{r}_{2})\rangle$ for a two electron atom using the relation attached in the image.I am trying to apply the formula for this case where $$m_{i}=1,m_{i'}=-1,m_{j}=1,m_{j'}=-1$$ and $$l_{i}=l_{i'}=l_{j}=l_{j'}=1$$

I really dont understand how to apply in the given formula as the summation is over all $m$'s. however, in the derivation it was clear that subscripted $l$'s and $m$'s are fixed and the one without subscript is varying. How do I apply this formula to evaluate these matrix elements ? Can someone suggest a better reference and help me in applying this formula

Source for this pdf :http://www.phys.ubbcluj.ro/~lnagy/pdf/1curs.pdf

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From Jackson's "Classical Electrodynamics", Eq. 3.70:

$$\frac{1}{r_{12}} = 4\pi \sum_{l=0}^\infty \sum_{m=-l}^l\frac{1}{2l+1}\frac{r_<^l}{r_>^{l+1}}Y^*_{lm}(\hat{r}_2)Y_{lm}(\hat{r}_1)$$

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  • $\begingroup$ I already know this expansion.. But I really dont know how to use formula in the attachment to evaluate the matrix elements. Can someone help me please ? $\endgroup$ – user135580 Nov 20 '18 at 12:42

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